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In the image, you provided a math exercise related to quadratic functions and their graphs. The tasks involve analyzing parabolas and determining key characteristics like vertex (Scheitelpunkt), stretch factor (Streckfaktor), and the equation of the functions. The first part of the exercise asks to identify these details for the functions \(f\), \(g\), \(h\), and \(i\), which are plotted on a graph.

### Task 5 Analysis:
- **Graph of Function \(f(x)\)** (red parabola):
  - You’ve identified that the vertex \(S\) is \((0, -2)\).
  - The stretch factor \(a\) is \(0.5\) (as noted).
  - The equation for the quadratic function is written in vertex form as:
    \[
    f(x) = 0.5 \cdot (x - 0)^2 + (-2)
    \]
    or simplified:
    \[
    f(x) = 0.5x^2 - 2
    \]
    
Next, you would apply the same analysis to the other functions \(g(x)\), \(h(x)\), and \(i(x)\), observing their vertex positions and stretch factors based on their appearance in the graph.

### Task 6: Intersection Points with the Axes
You are asked to find the intersection points (Schnittpunkte) of various functions with the coordinate axes.

Let’s go through **(a)** as an example:
1. \( f(x) = -1.5x^2 - 6 \)

   - **Intersection with the y-axis** occurs where \(x = 0\):
     \[
     f(0) = -1.5(0)^2 - 6 = -6
     \]
     So, the y-intercept is \( (0, -6) \).
     
   - **Intersection with the x-axis** occurs where \(f(x) = 0\):
     \[
     -1.5x^2 - 6 = 0
     \]
     Solving for \(x\):
     \[
     -1.5x^2 = 6 \quad \Rightarrow \quad x^2 = -4
     \]
     This has no real solution, so the function does not intersect the x-axis.

You would follow the same method for the other functions in parts (b) through (g), solving for their intersection points with the axes.

Would you like more detailed solutions for the remaining parts of Task 5 or Task 6?

### Related Questions:
1. How do you determine the vertex of a quadratic function from its equation?
2. What is the general formula for a parabola in vertex form, and how does the stretch factor affect it?
3. How do you find the x-intercepts and y-intercepts of a quadratic function?
4. Can every quadratic function intersect the x-axis, and under what conditions would it not?
5. How does changing the value of \(a\) (the stretch factor) affect the shape of a parabola?

### Tip:
To quickly find the vertex of a parabola given in standard form \(ax^2 + bx + c\), you can use the formula for the x-coordinate of the vertex: \(x = \frac{-b}{2a}\).

Determine the vertex and stretch factor for quadratic functions f, g, h, i, and k from their graphs. Also, find the intersection points of function f with the coordinate axes.

Learn how to determine the vertex and stretch factor of quadratic functions from their graphs and how to find the intersection points with coordinate axes. This problem covers key concepts such as the vertex form, quadratic equations, and graphing parabolas, suitable for Grades 9-11.

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